Abstract
In the last chapters we presented some of the most glamorous advances in Physics itself. Now we are ready to discuss some of the most intriguing among the countless contributions of the physical methodologies to other sciences. New approaches have been developed, for which the emergence of new concepts, patterns, universalities and the cross-fertilization among different phenomenologies, matter more than the field of application. Some of them, such as Complexity , have been very fruitful and brought along with them a new perspective on science and the underlying reality.
It used to be thought that the events that changed the world were things like big bombs, maniac politicians, huge earthquakes, or vast population movements, but it has now been realized that this is a very old-fashioned view held by people totally out of touch with modern thought. The things that really change the world, according to Chaos theory, are the tiny things. A butterfly flaps its wings in the Amazonian jungle, and subsequently a storm ravages half of Europe.
Neil Gaiman, Good Omens: The Nice and Accurate Prophecies of Agnes Nutter, Witch
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Notes
- 1.
Sunlight scatters off molecules of gas and other small particles in the atmosphere in a random process called Rayleigh scattering. It is this scattered light that gives the sky its brightness and colour. Since Rayleigh scattering is inversely proportional to the fourth power of wavelength, shorter wavelengths, such as violet and blue scatter more than longer ones, such as yellow and red. However, the whole process is much more complex, since this elementary explanation does not take into account other factors, such as the solar spectrum and the absorption properties of the Earth’s atmosphere.
- 2.
The Santa Fe Institute is an independent, nonprofit theoretical research institute located in New Mexico and dedicated to the multidisciplinary study of the fundamental principles of complex adaptive systems. It was founded in 1984 by Gell-Man and a few other scientists, mostly from the Los Alamos National Laboratory.
- 3.
See Chap. 3 for a discussion of modelling.
- 4.
Every high school student can solve second order algebraic equations. However, third order equations represented an unsolved enigma up to the early 16th century, when the mathematician Scipione del Ferro found a solution for a special case of them. At that time mathematical solutions of relevant problems were not usually published, since they allowed their owner to challenge other mathematicians and make money in highly publicised duels (in which problems were proposed and whoever could solve more of them would be the winner). Before he died, del Ferro confided his secret to his student, Antonio Fiore.
A young mathematician, Niccolò Tartaglia, challenged Fiore to a duel and unexpectedly won, gaining wide fame. But another famous mathematician, Girolamo Cardano, succeeded in inducing Tartaglia to reveal the formula to him by enticing him with many promises, one of which was to keep the secret. However, a few years later, Cardano published it, so that it has since been known as Cardano’s formula. The unfortunate Tartaglia challenged Cardano to a duel, which, however, took place against a student of Cardano, Lodovico Ferrari, who in the meantime had also learned to solve fourth order equations, and consequently won.
Centuries later, Niels Henrik Abel and Paolo Ruffini proved, in their impossibility theorem (1823), that there cannot be an analytical solution for fifth order equations. There is, however, nothing magical in this. In fact solution here means simply expression in terms of standard elementary functions, such as square roots, logarithms and trigonometric functions. However, in order to obtain a square root, one needs a numerical procedure with iterations, and an exact value is not possible, since, being an irrational number, it would require an infinite number of digits. Likewise, fifth order algebraic equations are also numerically soluble to any desired accuracy by means of iterations. Introducing new functions, besides the standard ones, it would be equally possible to express these solutions analytically.
- 5.
This image has been uploaded to Wikimedia Commons and made available under the Creative Com-mons Attribution-Share Alike 3.0 Unported license (https://commons.wikimedia.org/wiki/File:KochFlake.png).
- 6.
This image has been uploaded by António Miguel de Campos to Wikimedia Commons and made available under the Creative Com-mons Attribution-Share Alike 3.0 Unported license (https://commons.wikimedia.org/wiki/File:Peanocurve.svg).
- 7.
A conjecture is defined in the Merriam-Webster dictionary as an opinion or idea formed without proof or sufficient evidence. More specifically, in mathematics, as a proposition before it has been proved or disproved.
- 8.
Actually we would have preferred to use, instead of “major”, a word conveying our profound admiration, such as beautiful, but we have refrained, in order not to get involved in another trap, i.e. the necessity of its definition in our context.
- 9.
From greek: iso = same, and allo = different.
- 10.
This image has been made available under the Creative Commons Attribution-Share Alike 3.0 Unported license (https://commons.wikimedia.org/wiki/File:Kleiber1947.jpg).
- 11.
G.B. West has also been President of the Santa Fe Institute2 from 2005 to 2009.
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Barrett, R., Delsanto, P.P., Tartaglia, A. (2016). Complexity and Universality. In: Physics: The Ultimate Adventure. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-31691-8_12
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